366 REPOBT — 1880. 



o£ tlie other theorems of interpolation, the reader is referred to Mr. 

 Monlton's notes to ch. iii. of Boole's 'Finite Differences,' second edition, and* 

 to the examples of the same chapter. The extensions of Gauss's method 

 are of too special application to need exposition here, especially con- 

 sidering their very doubtful utility in dealing with actual, as distinguished 

 from analytical, data. 



It is important to observe that both Newton's and Cotes's methods 

 admit of a far wider generalisation of form. The parabolic character of 

 the assumption usually made, that the function subjected to interpolation 

 or quadrature is either a rational integral function of the variable, or a 

 convergent series, is not by any means necessary. On the contrary, the 

 very form of Lagrange's expression shows that it is permissible to sub- 

 stitute for the simple factors, functions of those factors arbitrarily chosen, 

 with only such restriction as to form as is needed to prevent the formula 

 becoming confused or nugatory.* This is merely another way of stating 

 the essentially indeterminate character of interpolation. It has to be 

 shown, as a prior condition of the use of any such specialised formula, 

 that there is good reason for applying it, and that its results are reliable. 

 The reason for generally selecting the ordinary methods turns upon the 

 two principles, that a. function can be approximately represented by a 

 convergent rational series, and that the approximation can be made as 

 great aswe please by taking the intervals sufficiently small. It has already 

 been pointed out that these principles are not universally true, and, as 

 particular cases, that they are not so when there is either physical dis- 

 continuity, or discontinuity within the meaning given to it in the proofs 

 of Taylor's theorem. An examination into the questions corresponding to 

 these is needed in using any such functional substitute for the simple 

 factors of the parabolic assumption, in order to render the method safe 

 and complete. There are doubtless many cases in which this may be 

 practically neglected. In those cases there may be a doubt as to the 

 necessity for any such refinement at all, except as a mere matter of 

 selecting the proper function for interpolation,! and indeed the complete 

 investigation frequently amounts, in the end, to nothing more than doing 

 this. It even sometimes brings back the question to the determination 

 of an analytical expression which shall adequately represent the table or 

 series of observations. A very remarkable instance of this is the repre- 

 sentation, due to the late Benjamin Gompertz,+ of the decrements of 

 life by means of a double exponential function. The equivalent 

 physical assumption is that the stock of vital force undergoes a weakening 

 proportional to the time, and this assumption, not improbable in itself, 

 is found, with a suitable determination of the parameters in each case, to 

 represent, with a high degree of accuracy, all the best life-tables, through 



* This may very well happen if very general forms are incautiously subjected ta 

 special interpretation, or if special forms are incautiously generalised. ^-' <\>x is a 

 well-known example of this trap for the unwary. 



t Cf . Stirling, Methodus Biff. p. 88, ' Nam interpolatio non est temere suscipienda, 

 sed ante exordium operis inquirendum est quajnam sit Series simplicissima, ex cujus 

 intercalatione pendet ea seriei propositre. Atque h:ec prteparatio est magna ex parte 

 omnino necessaria, ut deveniamus ad conclusiones concinnas et elegantes.' 



\ The formula is dy = — ah^ y dx, where y is the number living at the end of x 

 years. See Gompertz ' On the nature of the Function expressive of the law of 

 Human Mortality,' PMloi. Tram. 1826, p. 513. See also another paper by the same 

 author, ridl. Trans. 1862, p. 571. See also the article ' Mortality ' in the Penny 

 Cycloj)<Bdia and in the Enylish CycUjjcEdia (Arts and Sciences). 



